Lesson 16: Derivatives of Exponential and Logarithmic Functions

Section 3.3Derivatives of Exponential andLogarithmic FunctionsV63.0121, Calculus IMarch 10/11, 2009AnnouncementsQuiz 3 this week: Covers Sections 2.1–2.4Get half of all unearned ALEKS points by March 22Image credit: heipeiOutlineDerivative of the natural exponential functionExponential GrowthDerivative of the natural logarithm functionDerivatives of other exponentials and logarithmsOther exponentialsOther logarithmsLogarithmic DifferentiationThe power rule for irrational powersProof.Follow your nose:f(x + h) − f(x)ax+h − axf′(x) = lim= limh→0hh→0haxah − axah − 1= lim= ax · lim= ax · f′(0).h→0hh→0hTo reiterate: the derivative of an exponential function is a constanttimes that function. Much different from polynomials!Derivatives of Exponential FunctionsFactIf f(x) = ax, then f′(x) = f′(0)ax.To reiterate: the derivative of an exponential function is a constanttimes that function. Much different from polynomials!Derivatives of Exponential FunctionsFactIf f(x) = ax, then f′(x) = f′(0)ax.Proof.Follow your nose:f(x + h) − f(x)ax+h − axf′(x) = lim= limh→0hh→0haxah − axah − 1= lim= ax · lim= ax · f′(0).h→0hh→0hDerivatives of Exponential FunctionsFactIf f(x) = ax, then f′(x) = f′(0)ax.Proof.Follow your nose:f(x + h) − f(x)ax+h − axf′(x) = lim= limh→0hh→0haxah − axah − 1= lim= ax · lim= ax · f′(0).h→0hh→0hTo reiterate: the derivative of an exponential function is a constanttimes that function. Much different from polynomials!AnswerIf h is small enough, e ≈ (1 + h)1/h. So[]heh − 1(1 + h)1/h− 1≈(1 + h) − 1h=== 1hhhheh − 1So in the limit we get equality: lim= 1h→0hThe funny limit in the case of eRemember the definition of e:()1 ne = lim1 += lim (1 + h)1/hn→∞nh→0Questioneh − 1What is lim?h→0heh − 1So in the limit we get equality: lim= 1h→0hThe funny limit in the case of eRemember the definition of e:()1 ne = lim1 += lim (1 + h)1/hn→∞nh→0Questioneh − 1What is lim?h→0hAnswerIf h is small enough, e ≈ (1 + h)1/h. So[]heh − 1(1 + h)1/h− 1≈(1 + h) − 1h=== 1hhhhThe funny limit in the case of eRemember the definition of e:()1 ne = lim1 += lim (1 + h)1/hn→∞nh→0Questioneh − 1What is lim?h→0hAnswerIf h is small enough, e ≈ (1 + h)1/h. So[]heh − 1(1 + h)1/h− 1≈(1 + h) − 1h=== 1hhhheh − 1So in the limit we get equality: lim= 1h→0hDerivative of the natural exponential functionFrom()dah − 1eh − 1ax =limaxandlim= 1dxh→0hh→0hwe get:Theoremd ex = exdxExponential GrowthCommonly misused term to say something grows exponentiallyIt means the rate of change (derivative) is proportional to thecurrent valueExamples: Natural population growth, compounded interest,social networksDocument Outline

• Announcements
• Derivative of the natural exponential function
  • Exponential Growth
• Derivative of the natural logarithm function
• Derivatives of other exponentials and logarithms
  • Other exponentials
  • Other logarithms
• Logarithmic Differentiation
  • The power rule for irrational powers